Compactness Of Composition Operators Research Articles

Harmonic Bloch Function Spaces and their Composition Operators

In this paper we characterize some basic properties of composition operators on the spaces of harmonic Bloch functions. First we provide some equivalent conditions for boundedness and compactness of composition operators. In the sequel we investigate closed range composition operators. These results extends the similar results that were proven for composition operators on the Bloch spaces.

Counting functions for Dirichlet series and compactness of composition operators

AbstractWe give a sufficient condition for a composition operator with positive characteristic to be compact on the Hardy space of Dirichlet series.

On composition operators on the Wiener algebra of Dirichlet series

We show that the symbol of a bounded composition operator on the Wiener algebra of Dirichlet series does not need to belong to this algebra. Our example even gives an absolutely summing (hence compact) composition operator.

Composition operators from harmonic $ \mathcal{H}^{\infty} $ space into harmonic Zygmund space

<abstract><p>This research paper sought to characterize the boundedness and compactness of composition operators from the space $ \mathcal{H}^{\infty} $ of bounded harmonic mappings into harmonic Zygmund space $ \mathcal{Z}_H $, on the open unit disk. Furthermore, we obtain an estimate of the essential norms of such an operator. These results extends the similar results that were proven for composition operators on analytic function spaces.</p></abstract>

Linear connection between composition operators on the Hardy space

We consider the space of all composition operators, acting on the Hardy space over the unit disk, in the uniform operator topology. We obtain a characterization for linear connection between composition operators. As one of applications, we see that the set of all compact composition operators is a polygonally connected component, in sharp contrast to the known fact that this set is properly contained in a path connected component. When the inducing maps have “good” boundary behavior in the sense of higher-order data and order of contact, we extend/recover the Kriete-Moorhouse characterization for linear connection through a completely different approach relying on our results. We also notice some results in conjunction with the Bergman space case. Several questions motivated by our results are included.

Compactness of composition operators on the Bergman spaces of convex domains and analytic discs

We study the compactness of composition operators on the Bergman spaces of certain bounded convex domains in ℂn with non-trivial analytic discs contained in the boundary. As a consequence we characterize that compactness of the composition operator with a continuous symbol (up to the closure) on the Bergman space of the polydisc.

Compactification, and beyond, of composition operators on Hardy spaces by weights

We study when multiplication by a weight can turn a non-compact composition operator on \(H^2\) into a compact operator, and when it can be in Schatten classes. The \(q\)-summing case in \(H^p\) is considered. We also study when this multiplication can turn a compact composition operator into a non-compact one.

A mean counting function for Dirichlet series and compact composition operators

We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy–Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.

Weighted composition operators on spaces of functions with derivative in a Bergman space

Following the idea of the paper by Contreras and Hernandez-Diaz, we first give the characterization of the boundedness of the weighted composition operator $$W_{\varphi ,\phi }$$ on $$B^p$$ . Then, we investigate the boundedness and the compactness of composition operators $$C_{\varphi}$$ on $$B^p$$ and study the spectrum of the multiplication operator $$M_{\phi}$$ on $$B^p$$ , as well. Finally, motivated by the paper by Cuckovic and Paudyal, we describe the relationships between the invariant subspaces of $$M_{z}$$ on $$B^p_{0}$$ and T on $$A^p$$ , where T is the sum of multiplication operator and Volterra operator. Moreover, we provide some Beurling-type invariant subspaces of $$M_{z}$$ on $$B^p$$ and $$B^p_0$$ , respectively.

Composition Operators Between Weighted Fock Spaces

In this paper, we characterize completely the boundedness and compactness of composition operators between weighted Fock spaces.

Nevanlinna and Carleson functions of analytic self‐maps on multiply‐connected domains

AbstractIn the paper we prove that the Nevanlinna counting function and the Carleson function of analytic self‐maps of finitely connected domains with smooth boundary are equivalent. Using this result we obtain the characterization of compact composition operators on Hardy–Orlicz spaces.

Composition operators and closures of $\mathcal{Q}_K(p,q)$-type spaces in the Logarithmic Bloch space

Closures of $\mathcal{Q}_K(p,q)$-type spaces in the Logarithmic Bloch space are investigated in this paper. Moreover, we characterize the boundedness and compactness of composition operators from the Logarithmic Bloch space to the closure of $\mathcal{Q}_K(p,q)$ type spaces in the Logarithmic Bloch space

The Norm of a Composition Operator from the Space of Cauchy Transforms into Zygmund-Type Spaces

The problem of evaluation of the norm of a composition operator acting between the spaces of holomorphic functions is quite difficult. In the Hardy and weighted Bergman spaces, the norm of the composition operator is unknown even for the choice of a fairly simple symbol. We compute the operator norm of the composition operator acting between the space of Cauchy transforms and Zygmund-type spaces. We also characterize bounded and compact composition operators acting between these spaces.

Composition operators on Hardy spaces of the homogenous rooted trees

In Muthukumar and Ponnusamy (Bull Malays Math Sci Soc 40(4):1801–1815, 2017), the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space \(\mathbb {T}_{p}\) defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator \(C_\phi \) to be bounded on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p,0}\) spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on \({\mathbb {T}}_{p}\) and on \({\mathbb {T}}_{p,0}\) spaces. Also, we discuss the compactness of \(C_\phi \) on \({\mathbb {T}}_{p}\) and finally prove there are no compact composition operators on \({\mathbb {T}}_{p,0}\) spaces.

Composition Operators on Generalized Fock Spaces of Slice Hyperholomorphic Functions

In this paper, we study composition operators on generalized Fock spaces of slice hyperholomorphic functions. More precisely, some relationship for composition operators between the slice hyperholomorphic setting and the classical setting is explored to characterize the boundedness and compactness of composition operators. Motivated by a question in Kumar (Adv Appl Clifford Algebras 27:1459–1477, 2017), we characterize S-spectra of composition operators. Finally, we characterize connected components of the space of bounded composition operators.

Some Characterizations of Composition Operators on Weighted Dirichlet Spaces

In this paper, we give three different characterizations for the boundedness and compactness of composition operators between different weighted Dirichlet spaces in the unit disk.

Angular derivatives and compactness of composition operators on Hardy spaces

Let Do be a simply connected subdomain of the unit disk and A be a compact subset of Do. Let ϕ be a universal covering map for Do∖A. We prove that the composition operator Cϕ is compact on the Hardy space Hp if and only if ϕ does not have an angular derivative at any point of the unit circle. This result extends a theorem of M.M. Jones.

Composition Operators and the Closure of Morrey Space in the Bloch Space

In this paper, we characterize the closure of the Morrey space in the Bloch space. Furthermore, the boundedness and compactness of composition operators from the Bloch space to the closure of the Morrey space in the Bloch space are investigated.

Norm-attaining Composition Operators on Lipschitz Spaces

Every composition operator $C_{\varphi}$ on the Lipschitz space $\operatorname{Lip}_0(X)$ attains its norm. This fact is essentially known and we give in this paper a sequential characterization of the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{Lip}_0(X)$. We also characterize the norm-attaining composition operators $C_{\varphi}$ on the little Lipschitz space $\operatorname{lip}_0(X)$ which separates points uniformly and identify the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{lip}_0(X)$. We deduce that compact composition operators on $\operatorname{lip}_0(X)$ are norm-attaining whenever the sphere unit of $\operatorname{lip}_0(X)$ separates points uniformly. In particular, this condition is satisfied by spaces of little Lipschitz functions on Holder compact metric spaces $(X,d^{\alpha})$ with $0 \lt \alpha \lt 1$.

Composition Operators and Closures of Dirichlet Type Spaces Dμ in Bloch Type Spaces

We characterize the closure of Dirichlet type spaces Dμ in Bloch type spaces. Moreover, we study the boundedness and compactness of composition operators from Bloch type spaces to closures of Dirichlet type spaces Dμ in Bloch type spaces.